Characterization of Maximizers for Sums of the First Two Eigenvalues of Sturm-Liouville Operators

In this paper we study the maximization of the sum of the first two Dirichlet eigenvalues for Sturm-Liouville operators with potentials in the noncompact space L^1. We prove that there exists a unique potential function achieving the maximum, which is non-negative, piecewise smooth, and symmetric. Using measure differential equations and weak* convergence, we show that the nonzero part of the maximizer can be determined by the solution to the pendulum equation θ'' + ℓ sinθ = 0. This research has potential applications in areas such as quantum mechanics, vibration analysis, and structural engineering, where understanding eigenvalue optimization is crucial.